Is this true for any two matrices of complex numbers, and if not, what counterexamples are there? If one of these was a matrix of real numbers, would this property hold?

I've been playing around with this, but since I can't find a way to write $\det(A+B)$ in terms of $\det(A)$ and $\det(B)$. I haven't made much progress.

  • $\begingroup$ Regarding writing $\det(A+B)$ in terms of $\det(A),\det(B)$: linear algebra is not my strongest suit, but I believe that this is in general impossible. Certain approximations can be very useful, however, if the appropriate norm of one matrix is small with respect to the other. $\endgroup$ – FearfulSymmetry Sep 4 '20 at 18:35
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    $\begingroup$ @Kriesler, it is considered poor form on this website to delete or clear your question after you receive an answer. If you're satisfied with the answer provided, you can accept it. $\endgroup$ – FearfulSymmetry Sep 4 '20 at 19:00

No. Consider $A=\pmatrix{1&1\\ 0&1}$ and $B=\pmatrix{0&-1\\ i&0}$ for instance.


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